Academia.eduAcademia.edu

Figure from:Comparing Fuzzy, Probabilistic, and Possibilistic Partitions

See full PDFdownloadDownload figure
ig.4. Gaussians datasets used in Example 1. (a) Dataset 1. (b) Dataset 2.

Figure 9 ig.4. Gaussians datasets used in Example 1. (a) Dataset 1. (b) Dataset 2.

Related Figures (16)

Here, Nj, is the canonical (i.e., unit vector) basis of °. The 7th vertex of Nj, 1.€., In general, there are four class label types: crisp, fuzzy, proba- bilistic, and possibilistic. Let integer c be the number of classes, with 1 < c < n, and define three sets of Jabel vectors in RR as follows:
Here, Nj, is the canonical (i.e., unit vector) basis of °. The 7th vertex of Nj, 1.€., In general, there are four class label types: crisp, fuzzy, proba- bilistic, and possibilistic. Let integer c be the number of classes, with 1 < c < n, and define three sets of Jabel vectors in RR as follows:
CONTINGENCY TABLE AND FORMULAS USED TO COMPARE CRISP PARTITIONS U AND V TABLE I contains our conclusions and a discussion about conjectures and extensions of the new measures.
CONTINGENCY TABLE AND FORMULAS USED TO COMPARE CRISP PARTITIONS U AND V TABLE I contains our conclusions and a discussion about conjectures and extensions of the new measures.
Fig. 1. Four types for a pair of objects from O x O relative to clusters in U and V. (i) Paired in U and V. (ii) Neither paired in U nor in V. (iii) Paired in V but not in U. (iv) Paired in U but not in V.
Fig. 1. Four types for a pair of objects from O x O relative to clusters in U and V. (i) Paired in U and V. (ii) Neither paired in U nor in V. (iii) Paired in V but not in U. (iv) Paired in U but not in V.
Clusters U; and V, have two objects in common, i.e., { 0), 02}, U, and V, have 1 object in common, i.e., {03}, and U; and V3 have 0 objects in common. Additionally, there are two paired in U and V, ie., type (i), i.e., {{01, 02}, {01, 05} }, six in type (ii), ie., {{01, 04}, {01, 05}, {02, 04}, {02, 05 }, {03, 04}, {03, 05 }}, zero paired in type (iii), and two paired in type (iv), ie., {{o1, 03}, {02, 03}}. Equations (3a)-(3d), which are functions of the entries in Table I, are the building blocks of many similar- ity measures for s(U, V ). These four equations simply count the number of occurrences among the n(n — 1)/2 pairs of each of the four types of unordered pairs shown in Fig. |. Thus, a in (3a) is the number of pairs that are paired in some cluster of U and some cluster of V; b is the number of pairs neither paired in UV nor in V ,, etc. It is easy to verify that the calculation of the matrix N in Table I as N = UV" is correct; however, to our knowl- edge, it has never been exhibited quite this way. This simple observation will enable us to easily define soft generalizations of many of the comparison indices that are to be discussed next. Many well-known crisp comparison indices are functions of the values in Table I. Some of these indices have “canonical stories” attached to them that depend on interpretations of the next four functions. Here
Clusters U; and V, have two objects in common, i.e., { 0), 02}, U, and V, have 1 object in common, i.e., {03}, and U; and V3 have 0 objects in common. Additionally, there are two paired in U and V, ie., type (i), i.e., {{01, 02}, {01, 05} }, six in type (ii), ie., {{01, 04}, {01, 05}, {02, 04}, {02, 05 }, {03, 04}, {03, 05 }}, zero paired in type (iii), and two paired in type (iv), ie., {{o1, 03}, {02, 03}}. Equations (3a)-(3d), which are functions of the entries in Table I, are the building blocks of many similar- ity measures for s(U, V ). These four equations simply count the number of occurrences among the n(n — 1)/2 pairs of each of the four types of unordered pairs shown in Fig. |. Thus, a in (3a) is the number of pairs that are paired in some cluster of U and some cluster of V; b is the number of pairs neither paired in UV nor in V ,, etc. It is easy to verify that the calculation of the matrix N in Table I as N = UV" is correct; however, to our knowl- edge, it has never been exhibited quite this way. This simple observation will enable us to easily define soft generalizations of many of the comparison indices that are to be discussed next. Many well-known crisp comparison indices are functions of the values in Table I. Some of these indices have “canonical stories” attached to them that depend on interpretations of the next four functions. Here
Fig. 2. Comparison indices versus resubstitution error when r = c with ref. erence partition V.
Fig. 2. Comparison indices versus resubstitution error when r = c with ref. erence partition V.
The global solution of optimization problem (8) identifies the permutation of I, that is the best match to I,,. The solution to (8) grows with factorial complexity but can be solved by dynamic programming with, for example, the Kuhn—Munkres algorithm [42].
The global solution of optimization problem (8) identifies the permutation of I, that is the best match to I,,. The solution to (8) grows with factorial complexity but can be solved by dynamic programming with, for example, the Kuhn—Munkres algorithm [42].
Brouwer [37] discusses a different generalization to the three indices given in Table II: the Rand, namely, the Rand, a third variant of the adjusted Rand, and Jaccard’s index. Brouwer’s generalization of (3a) is given by In (12), cos(w!) is the angle between the vectors U? and U!, and cos(v}) is the angle between V} and V’. We see from (12) that the maximum occurs when both cosines are 1. Then, a = n(n — 1)/2, which will occur if and only if d = 0 in Rand’s formula, from which we conclude that U = V.
Brouwer [37] discusses a different generalization to the three indices given in Table II: the Rand, namely, the Rand, a third variant of the adjusted Rand, and Jaccard’s index. Brouwer’s generalization of (3a) is given by In (12), cos(w!) is the angle between the vectors U? and U!, and cos(v}) is the angle between V} and V’. We see from (12) that the maximum occurs when both cosines are 1. Then, a = n(n — 1)/2, which will occur if and only if d = 0 in Rand’s formula, from which we conclude that U = V.
Comparison of (15) with the fuzzy Rand indices of Campello (10), Hullermeier and Rifqi (11), and Brouwer (12) shows that the four indices are different from each other. The big ad- vantage of our method of forming the contingency table as N* = 6UV!? = [n/>>i_, nieJUV" is that this formulation di- rectly generalizes all 15 indices in Table II (and any other indices that are functions only of the entries in the contingency table) to every combination of (U,V). There are 16 possible pair types accordingly as each of U and V are crisp, fuzzy, probabilistic, or possibilistic; therefore, we have, for example, 16 Rand indices, 16 Jaccard indices, etc. Each formula there is recovered when U and V are crisp, i.e., these are true generalizations to every case—by definition.
Comparison of (15) with the fuzzy Rand indices of Campello (10), Hullermeier and Rifqi (11), and Brouwer (12) shows that the four indices are different from each other. The big ad- vantage of our method of forming the contingency table as N* = 6UV!? = [n/>>i_, nieJUV" is that this formulation di- rectly generalizes all 15 indices in Table II (and any other indices that are functions only of the entries in the contingency table) to every combination of (U,V). There are 16 possible pair types accordingly as each of U and V are crisp, fuzzy, probabilistic, or possibilistic; therefore, we have, for example, 16 Rand indices, 16 Jaccard indices, etc. Each formula there is recovered when U and V are crisp, i.e., these are true generalizations to every case—by definition.
Fig. 6. Original Rand for HCM and our soft generalizations, fuzzy and possibilistic Rand for FCM and PCM, respectively, for (a) dataset 1 and (b) dataset 2. Fig. 5. Terminal partitions U acquired by (a) HCM, (b) FCM, and (c) PCM for c = 4 on dataset 2.
Fig. 6. Original Rand for HCM and our soft generalizations, fuzzy and possibilistic Rand for FCM and PCM, respectively, for (a) dataset 1 and (b) dataset 2. Fig. 5. Terminal partitions U acquired by (a) HCM, (b) FCM, and (c) PCM for c = 4 on dataset 2.
Fig. 7. Comparison of the four fuzzy-Rand indices on FCM partitions of dataset 3, which has c = 5 clusters.
Fig. 7. Comparison of the four fuzzy-Rand indices on FCM partitions of dataset 3, which has c = 5 clusters.
Fig. 8. Comparison of the four fuzzy-Rand indices on EM partitions of dataset 3, which has c = 5 clusters.
Fig. 8. Comparison of the four fuzzy-Rand indices on EM partitions of dataset 3, which has c = 5 clusters.
RESULTS FOR THE 15 FORMULAS PROVIDED IN TABLE IT, DATASET 4, THE FCM, AND N* TABLE IV
RESULTS FOR THE 15 FORMULAS PROVIDED IN TABLE IT, DATASET 4, THE FCM, AND N* TABLE IV
COMPUTATIONAL COMPLEXITY FOR THE FIVE FUZZY-RAND INDICES neither, and together in either but not both). We compared our generalization of the classical Rand index with three other fuzzy generalizations of it both computationally, as well as in terms of computational complexity. We showed that our extension of the Rand index is O(n), while the other three are all O(n). Our numerical examples included the use of our soft comparison indices to validate both probabilistic and possibilistic partitions of the data. have valid fuzzy generalizations of all 15 formulas in Table IT. Moreover, as noted above, there are (many) other comparison indices that depend only on the entries of the contingency matrix N =UV’'; hence, our method is indeed quite general.
COMPUTATIONAL COMPLEXITY FOR THE FIVE FUZZY-RAND INDICES neither, and together in either but not both). We compared our generalization of the classical Rand index with three other fuzzy generalizations of it both computationally, as well as in terms of computational complexity. We showed that our extension of the Rand index is O(n), while the other three are all O(n). Our numerical examples included the use of our soft comparison indices to validate both probabilistic and possibilistic partitions of the data. have valid fuzzy generalizations of all 15 formulas in Table IT. Moreover, as noted above, there are (many) other comparison indices that depend only on the entries of the contingency matrix N =UV’'; hence, our method is indeed quite general.
a a a oe i Mr. Anderson received the Best Student Paper Award at the IEEE Interna- tional Conference on Fuzzy Systems at the World Congress on Computational Intelligence 2008.
a a a oe i Mr. Anderson received the Best Student Paper Award at the IEEE Interna- tional Conference on Fuzzy Systems at the World Congress on Computational Intelligence 2008.